Lusternik-Schnirelmann Category.

Format:

Book

Author/Creator:

Cornea, Octav.

Contributor:

Lupton, Gregory.

Oprea, John.

Series:

Mathematical Surveys and Monographs

Mathematical Surveys and Monographs ; v.103

Language:

English

Subjects (All):

Algebraic topology--Congresses.

Lusternik-Schnirelmann category--Congresses.

Local Subjects:

Algebraic topology--Congresses.

Lusternik-Schnirelmann category--Congresses.

Physical Description:

1 online resource (352 p.)

Other Title:

Mathematical Surveys and Monographs

Mathematical Surveys and Monographs, Volume 103

Place of Publication:

Providence : American Mathematical Society, 2014.

Language Note:

English

Summary:

''Lusternik-Schnirelmann category is like a Picasso painting. Looking at category from different perspectives produces completely different impressions of category's beauty and applicability.'' --from the Introduction Lusternik-Schnirelmann category is a subject with ties to both algebraic topology and dynamical systems. The authors take LS-category as the central theme, and then develop topics in topology and dynamics around it. Included are exercises and many examples. The book presents the material in a rich, expository style. The book provides a unified approach to LS-category, including foundational material on homotopy theoretic aspects, the Lusternik-Schnirelmann theorem on critical points, and more advanced topics such as Hopf invariants, the construction of functions with few critical points, connections with symplectic geometry, the complexity of algorithms, and category of $3$-manifolds. This is the first book to synthesize these topics. It takes readers from the very basics of the subject to the state of the art. Prerequisites are few: two semesters of algebraic topology and, perhaps, differential topology. It is suitable for graduate students and researchers interested

Contents:

""Contents""; ""Preface""; ""Chapter 1. Introduction to LS-Category""; ""1.1. Introduction""; ""1.2. The Definition and Basic Properties""; ""1.3. The Lusternik-Schnirelmann Theorem""; ""1.4. Sums, Homotopy Invariance and Mapping Cones""; ""1.5. Products and Fibrations""; ""1.6. The Whitehead and Ganea Formulations of Category""; ""1.7. Axioms and Category""; ""Exercises for Chapter 1""; ""Chapter 2. Lower Bounds for LS-Category""; ""2.1. Introduction""; ""2.2. Ganea Fibrations of a Product""; ""2.3. Toomer's Invariant""; ""2.4. Weak Category""; ""2.5. Conilpotency of a Suspension""

""2.6. Suspension of the Category""""2.7. Category Weight""; ""2.8. Comparison Theorem""; ""2.9. Examples""; ""Exercises for Chapter 2""; ""Chapter 3. Upper Bounds for Category""; ""3.1. Introduction""; ""3.2. First Properties of Upper Bounds""; ""3.3. Geometric Category is not a Homotopy Invariant""; ""3.4. Strong Category and Category Differ by at Most One""; ""3.5. Cone-length""; ""3.6. Stabilization of Ball Category""; ""3.7. Constraints Implying Equality of Category and Upper Bounds""; ""Exercises for Chapter 3""; ""Chapter 4. Localization and Category""; ""4.1. Introduction""

""4.2. Localization of Groups and Spaces""""4.3. Localization and Category""; ""4.4. Category and the Mislin Genus""; ""4.5. Fibrewise Construction""; ""4.6. Fibrewise Construction and Category""; ""4.7. Examples of Fibrewise Construction""; ""Exercises for Chapter 4""; ""Chapter 5. Rational Homotopy and Category""; ""5.1. Introduction""; ""5.2. Rational Homotopy Theory""; ""5.3. Rational Category and Minimal Models""; ""5.4. Rational Category and Fibrations, Including Products""; ""5.5. Lower and Upper Bounds in the Rational Context""; ""5.6. Geometric Version of meat""

""Exercises for Chapter 5""""Chapter 6. Hopf Invariants""; ""6.1. Introduction""; ""6.2. Hopf Invariants of Maps ...""; ""6.3. The Berstein-Hilton Definition""; ""6.4. Hopf Invariants and LS-category""; ""6.5. Crude Hopf Invariants""; ""6.6. Examples""; ""6.7. Hopf-Ganea Invariants""; ""6.8. Iwase's Counterexamples to the Ganea Conjecture""; ""6.9. Fibrewise Construction and Hopf Invariants""; ""Exercises for Chapter 6""; ""Chapter 7. Category and Critical Points""; ""7.1. Introduction""; ""7.2. Relative Category""; ""7.3. Local Study of Isolated Critical Points""

""7.4. Functions with Few Critical Points: the Stable Case""""7.5. Closed Manifolds""; ""7.6. Fusion of Critical Points and Hopf Invariants""; ""7.7. Functions Quadratic at Infinity""; ""Exercises for Chapter 7""; ""Chapter 8. Category and Symplectic Topology""; ""8.1. Introduction""; ""8.2. The Arnold Conjecture""; ""8.3. Manifolds with ...""; ""8.4. The Arnold Conjecture for Symplectically Aspherical Manifolds""; ""8.5. Other Symplectic Connections""; ""Exercises for Chapter 8""; ""Chapter 9. Examples, Computations and Extensions""; ""9.1. Introduction""

""9.2. Category and the Free Loop Space""

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 311-324) and index.

Description based on print version record.

ISBN:

1-4704-1330-2